6.02e Calculate KE and PE: using formulae

6. A particle \(P\) is projected from a fixed point \(A\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal and moves freely under gravity. As \(P\) passes through the point \(B\) on its path, \(P\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\beta\) below the horizontal.

1. By considering energy, find the vertical distance between \(A\) and \(B\). Particle \(P\) takes 1.5 seconds to travel from \(A\) to \(B\).
2. Find the size of angle \(\alpha\).
3. Find the size of angle \(\beta\).
4. Find the length of time for which the speed of \(P\) is less than \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6. \begin{figure}[h] \end{figure} A rough straight ramp is fixed to horizontal ground. The ramp has length 15 m and is inclined at an angle \(\alpha\) to the ground, where \(\tan \alpha = \frac { 5 } { 12 }\). The line \(A B\) is a line of greatest slope of the ramp, where \(A\) is at the bottom of the ramp, and \(B\) is at the top of the ramp, as shown in Figure 3. A particle \(P\) of mass 6 kg is projected up the ramp with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) in a straight line towards \(B\). The coefficient of friction between \(P\) and the ramp is 0.25

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\). At the instant \(P\) reaches \(B\), the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). After leaving the ramp at \(B\), the particle \(P\) moves freely under gravity until it hits the horizontal ground at the point \(C\). Immediately before hitting the ground at \(C\), the speed of \(P\) is \(w \mathrm {~ms} ^ { - 1 }\)
2. Use the work-energy principle to find
   1. the value of \(v\),
   2. the value of \(w\).
      
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1. \hspace{0pt} [In this question, the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane with \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]

\begin{figure}[h] \end{figure} A small ball is projected with velocity \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) from the fixed point \(A\).
The point \(A\) is 20 m above horizontal ground.
The ball hits the ground at the point \(B\), as shown in Figure 4.
The ball is modelled as a particle moving freely under gravity.

1. By considering energy, find the speed of the ball at the instant immediately before it hits the ground.
2. Find the direction of motion of the ball at the instant immediately before it hits the ground.
3. Find the time taken for the ball to travel from \(A\) to \(B\). At the instant when the direction of motion of the ball is perpendicular to ( \(3 \mathbf { i } + 2 \mathbf { j }\) ) the ball is \(h\) metres above the ground.
4. Find the value of \(h\).

7. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards. Position vectors are given relative to a fixed origin O.] At time \(t = 0\) seconds, the particle \(P\) is projected from \(O\) with velocity ( \(3 \mathbf { i } + \lambda \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(\lambda\) is a positive constant. The particle moves freely under gravity. As \(P\) passes through the fixed point \(A\) it has velocity \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The kinetic energy of \(P\) at the instant it passes through \(A\) is half the initial kinetic energy of \(P\). Find the position vector of \(A\), giving the components to 2 significant figures.
(10)

7. \begin{figure}[h] \end{figure} A small ball \(P\) is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is 47.5 m above a horizontal beach. The ball moves freely under gravity and hits the beach at the point \(B\), as shown in Figure 3.

1. By considering energy, find the speed of \(P\) immediately before it hits the beach. The ball was projected from \(A\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\)
2. Find the greatest height above the beach of \(P\) as it moved from \(A\) to \(B\).
3. Find the least speed of \(P\) as it moved between \(A\) and \(B\).
4. Find the horizontal distance from \(A\) to \(B\).

8. \begin{figure}[h] \end{figure} The fixed point \(A\) is \(h\) metres vertically above the point \(O\) that is on horizontal ground. At time \(t = 0\), a particle \(P\) is projected from \(A\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle moves freely under gravity. At time \(t = 2.5\) seconds, \(P\) strikes the ground at the point \(B\). At the instant when \(P\) strikes the ground, the speed of \(P\) is \(18 \mathrm {~ms} ^ { - 1 }\), as shown in Figure 4.

1. By considering energy, find the value of \(h\).
2. Find the distance \(O B\). As \(P\) moves from \(A\) to \(B\), the speed of \(P\) is less than or equal to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T\) seconds.
3. Find the value of \(T\)

7. \begin{figure}[h] \end{figure} At time \(t = 0\), a particle \(P\) of mass 0.7 kg is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) at an angle \(\theta ^ { \circ }\) to the horizontal. The particle moves freely under gravity. At time \(t = 2\) seconds, \(P\) passes through the point \(A\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving downwards at \(45 ^ { \circ }\) to the horizontal, as shown in Figure 4. Find

1. the value of \(\theta\),
2. the kinetic energy of \(P\) as it reaches the highest point of its path. For an interval of \(T\) seconds, the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is such that \(v \leqslant 6\)
3. Find the value of \(T\).

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5. A particle \(P\) is projected with velocity \(( 2 u \mathbf { i } + 3 u \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) from a point \(O\) on a horizontal plane, where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical unit vectors respectively. The particle \(P\) strikes the plane at the point \(A\) which is 735 m from \(O\).

1. Show that \(u = 24.5\).
2. Find the time of flight from \(O\) to \(A\). The particle \(P\) passes through a point \(B\) with speed \(65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the height of \(B\) above the horizontal plane.

4. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string. The particle \(A\) lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The particle \(B\) hangs freely below \(P\), as shown in Figure 2. The particles are released from rest with the string taut and the section of the string from \(A\) to \(P\) parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 5 } { 8 }\). When each particle has moved a distance \(h , B\) has not reached the ground and \(A\) has not reached \(P\).

1. Find an expression for the potential energy lost by the system when each particle has moved a distance \(h\). When each particle has moved a distance \(h\), they are moving with speed \(v\). Using the workenergy principle,
2. find an expression for \(v ^ { 2 }\), giving your answer in the form \(k g h\), where \(k\) is a number.

6. \begin{figure}[h] \end{figure} A golf ball \(P\) is projected with speed \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) on a cliff above horizontal ground. The angle of projection is \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\). The ball moves freely under gravity and hits the ground at the point \(B\), as shown in Figure 4.

1. Find the greatest height of \(P\) above the level of \(A\). The horizontal distance from \(A\) to \(B\) is 168 m .
2. Find the height of \(A\) above the ground. By considering energy, or otherwise,
3. find the speed of \(P\) as it hits the ground at \(B\).

3. \begin{figure}[h] \end{figure} A package of mass 3.5 kg is sliding down a ramp. The package is modelled as a particle and the ramp as a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The package slides down a line of greatest slope of the plane from a point \(A\) to a point \(B\), where \(A B = 14 \mathrm {~m}\). At \(A\) the package has speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(B\) the package has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 1. Find

1. the total energy lost by the package in travelling from \(A\) to \(B\),
2. the coefficient of friction between the package and the ramp.

7. A particle, of mass 0.3 kg , is projected from a point \(O\) on horizontal ground with speed \(u\). The particle is projected at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = 2\), and moves freely under gravity. When the particle has moved a horizontal distance \(x\) from \(O\), its height above the ground is \(y\).

1. Show that $$y = 2 x - \frac { 5 g } { 2 u ^ { 2 } } x ^ { 2 }$$ The particle hits the ground at the point \(A\), where \(O A = 36 \mathrm {~m}\).
2. Find \(u\), the speed of projection.
3. Find the minimum kinetic energy of the particle as it moves between \(O\) and \(A\). The point \(B\) lies on the path of the particle. The direction of motion of the particle at \(B\) is perpendicular to the initial direction of motion of the particle.
4. Find the horizontal distance between \(O\) and \(B\).

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is held at the point \(A\), where \(O A = l\) and \(O A\) is horizontal. The particle is then projected vertically downwards from \(A\) with speed \(\sqrt { 2 g l }\), as shown in Figure 4 . When the string makes an angle \(\theta\) with the downward vertical through \(O\) and the string is still taut, the tension in the string is \(T\).

1. Show that \(T = m g ( 3 \cos \theta + 2 )\) At the instant when the particle reaches the point \(B\), the string becomes slack.
2. Find the speed of \(P\) at \(B\).
3. Find the greatest height above \(O\) reached by \(P\) in the subsequent motion.

2. A particle \(P\) of mass \(m\) is at a distance \(x\) above the surface of the Earth. The Earth exerts a gravitational force on \(P\). This force is directed towards the centre of the Earth. The magnitude of this force is inversely proportional to the square of the distance of \(P\) from the centre of the Earth. At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a fixed sphere of radius \(R\).

1. Show that the magnitude of the gravitational force on \(P\) is \(\frac { m g R ^ { 2 } } { ( x + R ) ^ { 2 } }\) A particle is released from rest from a point above the surface of the Earth. When the particle is at a distance \(R\) above the surface of the Earth, the particle has speed \(U\). Air resistance is modelled as being negligible.
2. Find, in terms of \(U , g\) and \(R\), the speed of the particle when it strikes the surface of the Earth.
   

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6. \begin{figure}[h] \end{figure} A particle of mass \(m\) is attached to one end of a light inextensible string of length \(2 a\). The other end of the string is attached to a fixed point \(O\). The particle is initially held at the point \(A\) with the string taut and \(O A\) making an angle of \(60 ^ { \circ }\) with the downward vertical. The particle is then projected upwards with a speed of \(3 \sqrt { a g }\), perpendicular to \(O A\), in the vertical plane containing \(O A\), as shown in Figure 7. In an initial model of the motion of the particle, it is assumed that the string does not break. Using this model,

1. show that the particle performs complete vertical circles. In a refined model it is assumed that the string will break if the tension in it exceeds 7 mg . Using this refined model,
2. show that the particle still performs complete vertical circles. \includegraphics[max width=\textwidth, alt={}, center]{8a687d17-ec7e-463f-84dd-605f5c230db1-20_2249_50_314_1982}

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can rotate freely in a vertical plane about \(O\). The particle is projected with speed \(u\) from a point \(A\). The line \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\), where \(\alpha < \frac { \pi } { 2 }\) When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\), the speed of \(P\) is \(v\), as shown in Figure 4.

1. Show that \(v ^ { 2 } = u ^ { 2 } - 2 g l ( \cos \theta - \cos \alpha )\) Given that \(\cos \alpha = \frac { 2 } { 5 }\) and that \(u = \sqrt { 3 g l }\)
2. show that \(P\) moves in a complete vertical circle. As the rod rotates, the least tension in the rod is \(T\) and the greatest tension is \(k T\)
3. Find the exact value of \(k\)

6. \begin{figure}[h] \end{figure} A smooth hollow cylinder of internal radius \(a\) is fixed with its axis horizontal. A particle \(P\) moves on the inner surface of the cylinder in a vertical circle with radius \(a\) and centre \(O\), where \(O\) lies on the axis of the cylinder. The particle is projected vertically downwards with speed \(u\) from point \(A\) on the circle, where \(O A\) is horizontal. The particle first loses contact with the cylinder at the point \(B\), where \(\angle A O B = 150 ^ { \circ }\), as shown in Figure 3. Given that air resistance can be ignored,

1. show that the speed of \(P\) at \(B\) is \(\sqrt { } \left( \frac { a g } { 2 } \right)\),
2. find \(u\) in terms of \(a\) and \(g\). After losing contact with the cylinder, \(P\) crosses the diameter through \(A\) at the point \(D\). At \(D\) the velocity of \(P\) makes an angle \(\theta ^ { \circ }\) with the horizontal.
3. Find the value of \(\theta\).

4. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 4 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest at the point \(A\) on a rough fixed plane inclined at \(\alpha\) to the horizontal ground, where \(\sin \alpha = \frac { 3 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley and 2.5 m above the ground, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. The system is released from rest with the string taut. At the instant when \(Q\) hits the ground, \(P\) is at the point \(B\) on the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\).

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Find the total potential energy lost by the system as \(P\) moves from \(A\) to \(B\).
3. Find, using the work-energy principle, the speed of \(P\) as it passes through \(B\).

7. \begin{figure}[h] \end{figure} At time \(t = 0\), a particle \(P\) of mass 0.7 kg is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) at an angle \(\theta ^ { \circ }\) to the horizontal. The particle moves freely under gravity. At time \(t = 2\) seconds, \(P\) passes through the point \(A\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving downwards at \(45 ^ { \circ }\) to the horizontal, as shown in Figure 4. Find

1. the value of \(\theta\),
2. the kinetic energy of \(P\) as it reaches the highest point of its path. For an interval of \(T\) seconds, the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is such that \(v \leqslant 6\)
3. Find the value of \(T\).

5 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-3_291_182_799_945} Particles \(P\) and \(Q\), of masses 0.4 kg and \(m \mathrm {~kg}\) respectively, are joined by a light inextensible string which passes over a smooth pulley. The particles are released from rest at the same height above a horizontal surface; the string is taut and the portions of the string not in contact with the pulley are vertical (see diagram). \(Q\) begins to descend with acceleration \(2.45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and reaches the surface 0.3 s after being released. Subsequently, \(Q\) remains at rest and \(P\) never reaches the pulley.

1. Calculate the tension in the string while \(Q\) is in motion.
2. Calculate the momentum lost by \(Q\) when it reaches the surface.
3. Calculate the greatest height of \(P\) above the surface. \section*{[Questions 6 and 7 are printed overleaf.]}

2 \includegraphics[max width=\textwidth, alt={}, center]{3daca234-9b7f-41d4-bbaa-d35615a120fc-2_510_755_667_696} A uniform circular disc with centre \(A\) has mass \(M\) and radius \(3 a\). A second uniform circular disc with centre \(B\) has mass \(\frac { 1 } { 9 } M\) and radius \(a\). The two discs are rigidly joined together so that they lie in the same plane with their circumferences touching. The line of centres meets the circumference of the larger disc at \(P\) and the circumference of the smaller disc at \(O\). A particle of mass \(\frac { 1 } { 3 } M\) is attached at \(P\) (see diagram). Show that the moment of inertia of the system about an axis through \(O\), perpendicular to the plane of the discs, is \(51 M a ^ { 2 }\). The system is free to rotate about a fixed horizontal axis through \(O\), perpendicular to the plane of the discs. The system is held with \(O P\) horizontal and is then released from rest. Given that \(a = 0.5 \mathrm {~m}\), find the greatest speed of \(P\) in the subsequent motion, giving your answer correct to 2 significant figures.
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One end of a light inextensible string of length \(\frac { 3 } { 2 } a\) is attached to a fixed point \(O\) on a horizontal surface. The other end of the string is attached to a particle \(P\) of mass \(m\). The string passes over a small fixed smooth peg \(A\) which is at a distance \(a\) vertically above \(O\). The system is in equilibrium with \(P\) hanging vertically below \(A\) and the string taut. The particle is projected horizontally with speed \(u\) (see diagram). When \(P\) is at the same horizontal level as \(A\), the tension in the string is \(T\). Show that \(T = \frac { 2 m } { a } \left( u ^ { 2 } - a g \right)\). The ratio of the tensions in the string immediately before, and immediately after, the string loses contact with the peg is \(5 : 1\).

1. Show that \(u ^ { 2 } = 5 a g\).
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is next at the same horizontal level as \(A\).

4 A particle \(P\) is at rest at the lowest point on the smooth inner surface of a hollow sphere with centre \(O\) and radius \(a\). The particle is projected horizontally with speed \(u\) and begins to move in a vertical circle on the inner surface of the sphere. The particle loses contact with the sphere at the point \(A\), where \(O A\) makes an angle \(\theta\) with the upward vertical through \(O\). Given that the speed of \(P\) at \(A\) is \(\sqrt { } \left( \frac { 3 } { 5 } a g \right)\), find \(u\) in terms of \(a\) and \(g\). Find, in terms of \(a\), the greatest height above the level of \(O\) achieved by \(P\) in its subsequent motion. (You may assume that \(P\) achieves its greatest height before it makes any further contact with the sphere.)

A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).

1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).

A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).

1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).